Manuscript II-B@1 · II-B-v1.tex · remark · bfd2eb3c

Remark: The fit is not a mixture of instruments

An exact finite response-table instrument › Complete binary instruments and centered response · lines 1797–1817

II-B@1#rem:not-convex

Equation (eq:operational-fit) does not assert

B_{m} = \frac{2}{3} B_{n_{ℓ_{a}}} + \frac{2}{3} B_{n_{ℓ_{b}}} .

The right side of this false equation has total trace $4/3$ . The correct identity is the centered one, equivalently

B_{m} = \frac{2}{3} B_{n_{ℓ_{a}}} + \frac{2}{3} B_{n_{ℓ_{b}}} - \frac{1}{3} B_{0} .

This is an affine linear identity, not an instruction to implement a negative-probability mixture. The physical instrument $B_{m}$ is implemented directly by its own declared controlled gates.

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unlabeled · line 1800
$B_{m} = \frac{2}{3} B_{n_{ℓ_{a}}} + \frac{2}{3} B_{n_{ℓ_{b}}} .$
unlabeled · line 1807
$B_{m} = \frac{2}{3} B_{n_{ℓ_{a}}} + \frac{2}{3} B_{n_{ℓ_{b}}} - \frac{1}{3} B_{0} .$

Remark: The fit is not a mixture of instruments

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